Next, to obtain Theorem 2 of [1]}, in (REF ) we set \(\mathrm {F}^{\textup {Env}} \left(a,b\right)=\mathrm {F}^{\textup {Task}} \left(a,b\right)=(a-b)\) , \(\theta _{\textup {tsk}}=\beta \) , and \(\theta _{\textup {env}}=\lambda \) , and these choices leads to \(k_t=k_e=1\)  and \(\mathrm {G}_{\textup {Task}} \left(c\right)=\mathrm {G}_{\textup {Env}} \left(c\right)=c\) .
To simplify
the log-term, since the prior in independent of the data, by interchanging the order of expectations over \(\mathrm {P}_{T_{1:}}\mathrm {P}_{ \mathbf {Z}_{1:}^{\mathrm {M}}|T_{1:}}\)  and priors, and
recalling that the loss function is bounded
on the interval \([0,1]\) , and hence it is sub-Gaussian with parameter \(\sigma =0.5\) , we can conclude
\(\mathbb {E}_{U\sim \mathcal {Q}}\left(\mathrm {L}_{\mathrm {P}_{T\mathbf {Z}^\mathrm {M}}}(U)\right)-\mathbb {E}_{U\sim \mathcal {Q}} \mathrm {L}_{\mathbf {Z}_{1:}^{\mathrm {M}}}(U)\le \min _{\beta ,\lambda \ge 0}(\frac{\lambda }{8}+\frac{\lambda }{8\mathrm {M}})-\frac{1}{\sqrt{}}\log \delta \nonumber \\+ \frac{1}{\beta }\mathbb {E}_{\mathcal {Q}}\left(\sum _{i=1}^{}\mathrm {D}\left(\mathrm {Q}_i||\mathrm {P}\right)\right)+ \left( \frac{1}{\beta }+\frac{1}{\lambda }\right)\mathrm {D}\left(\mathcal {Q}||\mathcal {P}\right),\) 
